1. Field of the Invention
The present invention relates to a method of designing a magnetic field gradient coil assembly used in a nuclear magnetic resonance (NMR) spectrometer or the like and to a magnetic field gradient coil assembly designed by such a method.
2. Description of the Related Art
NMR spectrometers have adopted self-shielding magnetic field gradient coils because quickly rising magnetic field gradient pulses are applied to specimens. The designing procedures used heretofore are now summarized by referring to FIGS. 1–4. The designing procedures use the target field method as described by R. Tuner in A Target Field Approach to Optimal Coil Design, J. Phys. D: Appl. Phys. 19, L147–L151 (1986) and by P. Mansfield and B. Chapman in Multishield Active Magnetic Screening of Coil Structures in NMR, J. Magn. Reso. 72, pp. 211–223 (1987). In the target field method, the distributions of electric currents flowing through virtual inside and outside coils are calculated. Furthermore, more coils are distributed computationally. In this way, current distributions are approximated.
FIG. 1 is a schematic diagram of an NMR detector in which a self-shielding magnetic field gradient coil is mounted. FIG. 2 is a schematic view illustrating a system of coordinates used for designing and calculations, as well as a coil bobbin. FIG. 3(a) is a schematic view showing an inner coil. FIG. 3(b) is a schematic view showing an outer coil. These inner and outer coils together form a Z-axis magnetic field gradient coil assembly. FIG. 4 is a flowchart illustrating calculational procedures. As shown in FIG. 1, the NMR detector is comprised of a cover 01, the aforementioned inner coil, indicated by numeral 02, the aforementioned outer coil, indicated by numeral 03, and an NMR detection coil 04. This detector has the coil bobbin in which cylinders 51, 52, 53, and 54 are formed, as shown in FIG. 2. These cylinders 51, 52, 53, and 54 are also referred to as cylinders 1–4, respectively, and have radii of rt, rp, rs, and rb, respectively. The radius rt of cylinder 1 is the radius of a target magnetic field that takes the form of the cylinder 51. The radius rp is the radius of the primary, inner coil, or the cylinder 52. The radius rs is the radius of the outer screen coil, or the cylinder 53. The radius rp is the radius of the boundary defined by the cylinder 54.
As an example, a Z-axis magnetic field gradient coil assembly is designed using the inner and outer coils in such a way that the magnetic field strength at the boundary having the radius rb is 0 (self-shielding) and that a target magnetic field strength is obtained at the radius rt at which a specimen is placed. A Green's function G(k) used for magnetic field analysis is defined by Eqs. (1)–(6) below.
                                          B            ⁡                          (              z              )                                            p            ,                          r              =                              r                t                                                    =                              μ                          2              ⁢              π                                ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          ⅇ                                  ⅈ                  ⁢                                                                          ⁢                  kz                                            ⁢                                                j                  ⁡                                      (                    k                    )                                                  p                            ⁢                                                                    G                    ⁡                                          (                      k                      )                                                        ⁢                                      ⅆ                    k                                                                                        r                    p                                    ->                                      r                    t                                                                                                          Eq        .                                  ⁢        1                                                      B            ⁡                          (              z              )                                            s            ,                          r              =                              r                t                                                    =                              μ                          2              ⁢              π                                ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          ⅇ                                  ⅈ                  ⁢                                                                          ⁢                  kz                                            ⁢                                                j                  ⁡                                      (                    k                    )                                                  s                            ⁢                                                                    G                    ⁡                                          (                      k                      )                                                        ⁢                                      ⅆ                    k                                                                                        r                    s                                    ->                                      r                    t                                                                                                          Eq        .                                  ⁢        2                                                      B            ⁡                          (              z              )                                            p            ,                          r              =                              r                b                                                    =                              μ                          2              ⁢              π                                ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          ⅇ                                  ⅈ                  ⁢                                                                          ⁢                  kz                                            ⁢                                                j                  ⁡                                      (                    k                    )                                                  p                            ⁢                                                G                  ⁢                                      (                    k                    )                                    ⁢                                      ⅆ                    k                                                                                        r                    p                                    ->                                      r                    b                                                                                                          Eq        .                                  ⁢                  (          3          )                                                              B            ⁡                          (              z              )                                            s            ,                          r              =                              r                b                                                    =                              μ                          2              ⁢              π                                ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          ⅇ                                  ⅈ                  ⁢                                                                          ⁢                  kz                                            ⁢                                                j                  ⁡                                      (                    k                    )                                                  s                            ⁢                                                G                  ⁢                                      (                    k                    )                                    ⁢                                      ⅆ                    k                                                                                        r                    s                                    ->                                      r                    b                                                                                                          Eq        .                                  ⁢                  (          4          )                                                              j            ⁡                          (              k              )                                p                ,                                                            j                ⁡                                  (                  k                  )                                            s                        :                                          j                ⁡                                  (                  k                  )                                            s                                =                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          ⅇ                                                      -                    ⅈ                                    ⁢                                                                          ⁢                  kz                                            ⁢                              J                ⁡                                  (                  z                  )                                            ⁢                                                          ⁢                              ⅆ                z                                                                        Eq        .                                  ⁢                  (          5          )                                                              G            ⁡                          (              k              )                                                          r              p                        ->                          r              t                                      ,                              G            ⁡                          (              k              )                                                          r              s                        ->                          r              t                                      ,                              G            ⁡                          (              k              )                                                          r              p                        ->                          r              b                                      ,                              G            ⁡                          (              k              )                                                          r              s                        ->                          r              b                                                          Esq        .                                  ⁢                  (          6          )                    
Eq. (1) indicates a magnetic field produced by the inner coil subassembly at the radius rt. Eq. (2) indicates a magnetic field by the outer coil subassembly at the radius rt. Eq. (3) indicates a magnetic field developed by the inner coil subassembly at the boundary radius rb obtained by calculation. Eq. (4) indicates a magnetic field set up by the outer coil subassembly at the boundary radius rb obtained by calculation. Eq. (5) defines Fourier components j(k)p and j(k)s of the current density distributions in the inner and outer coil subassemblies, respectively. Eq. (6) indicates a current-field Green's function (response function) in a Fourier space. In this equation, rp→rt, rs→rt, rp→rb, and rs→rb meant the inner and outer coils produce magnetic fields at the radii rt and rb, respectively.
The calculational procedures are next described by referring to the flowchart of FIG. 4. A target magnetic field distribution TARGET(z) is set. A target value of the magnetic field distribution generated by the coil assembly near the center axis of the cylinder of the detector is set (step S1). The Fourier components target(k) of the target magnetic field distribution TARGET(z) are calculated using Eq. (7) (step S2).
                              target          ⁡                      (            k            )                          =                              ∫                          -              ∞                                      +              ∞                                ⁢                                    ⅇ                                                -                  ⅈ                                ⁢                                                                  ⁢                kz                                      ⁢                                                  ⁢                          TARGET              ⁡                              (                z                )                                      ⁢                          ⅆ              z                                                          Eq        .                                  ⁢                  (          7          )                    
A current-field Green function (response function) adapted for the designed coil assembly is found. Using this Green's function and target(k), the Fourier components j(k)p and j(k)s of the distributions of the currents flowing through the virtual inner and outer self-shielding coils are calculated (step S3).
Where the sum of magnetic fields given by the Fourier components of the distributions of the currents flowing through the inner and outer coils at the boundary rb is null, shielding conditions are satisfied. Under this state, if the Fourier components j(k)p of the distribution of the current flowing through the inner coils is used, the Fourier components j(k)s of the distribution of the current flowing through the outer coils is given by Eq. (8).
                                          j            ⁡                          (              k              )                                s                =                              -                                          j                ⁡                                  (                  k                  )                                            p                                ⁢                                                    G                ⁡                                  (                  k                  )                                                                              r                  p                                ->                                  r                  b                                                                                    G                ⁡                                  (                  k                  )                                                                              r                  s                                ->                                  r                  b                                                                                        Eq        .                                  ⁢                  (          8          )                    
Accordingly, the Fourier components b(k) of the magnetic fields produced by the inner and outer coils are given by Eq. (9).
                                          b            ⁡                          (              k              )                                            r            =                          r              t                                      =                                                                              j                  ⁡                                      (                    k                    )                                                  p                            ⁢                                                G                  ⁡                                      (                    k                    )                                                                                        r                    p                                    ->                                      r                    t                                                                        +                                                            j                  ⁡                                      (                    k                    )                                                  s                            ⁢                                                G                  ⁡                                      (                    k                    )                                                                                        r                    s                                    ->                                      r                    t                                                                                =                                                    j                ⁡                                  (                  k                  )                                            p                        ⁢                          (                                                                    G                    ⁡                                          (                      k                      )                                                                                                  r                      p                                        ->                                          r                      t                                                                      -                                                                                                    G                        ⁡                                                  (                          k                          )                                                                                                                      r                          s                                                ->                                                  r                          t                                                                                      ⁢                                                                  G                        ⁡                                                  (                          k                          )                                                                                                                      r                          p                                                ->                                                  r                          b                                                                                                                                                G                      ⁡                                              (                        k                        )                                                                                                            r                        s                                            ->                                              r                        b                                                                                                        )                                                          Eq        .                                  ⁢                  (          9          )                    Since b(k)=target(k), the Fourier components j(k)p and j(k)s of the distributions of the currents flowing through the virtual inner and outer coils are respectively given by:
                                          j            ⁡                          (              k              )                                p                =                              target            ⁡                          (              k              )                                            [                                                            G                  ⁡                                      (                    k                    )                                                                                        r                    p                                    ->                                      r                    t                                                              -                                                                                          G                      ⁡                                              (                        k                        )                                                                                                            r                        s                                            ->                                              r                        t                                                                              ⁢                                                            G                      ⁡                                              (                        k                        )                                                                                                            r                        p                                            ->                                              r                        b                                                                                                                                  G                    ⁡                                          (                      k                      )                                                                                                  r                      s                                        ->                                          r                      b                                                                                            ]                                              Eq        .                                  ⁢                  (          10          )                                                              j            ⁡                          (              k              )                                s                =                              -                                          j                ⁡                                  (                  k                  )                                            p                                ⁢                                                    G                ⁡                                  (                  k                  )                                                                              r                  p                                ->                                  r                  b                                                                                    G                ⁡                                  (                  k                  )                                                                              r                  s                                ->                                  r                  b                                                                                        Eq        .                                  ⁢                  (          11          )                    
By inverse-Fourier-transforming the Fourier components j(k)p and j(k)s, the distributions J(z)p and J(z)s of the currents flowing through the virtual inner and outer coils are found (step S4). The distributions J(z)p and J(z)s of the is currents flowing through the virtual inner and outer coils are integrated, and the positions of the coils conforming to the calculated current distributions are determined (step S5).
With the above-described designing procedures, the currents flowing through the inner and outer coils are calculated simultaneously. Therefore, if the results of the calculations of the current distributions are ideal, and if the coils are large, then it is possible to wind the coils exactly. However, as the coils decrease in size, it becomes more difficult to fabricate the coils accurately, that is, it is necessary to wind conductive wire into a bobbin shape (i.e., distributed winding) according to the calculated coil geometry in order to obtain the current distributions found by calculations. Generally, many turns of wire are necessary to conform with the calculated coil geometry. Adjacent turns of wire come too close. Hence, it is difficult to wind the wire completely in accordance with the coil geometry. If lap winding is done, the coil diameter increases, thus varying the calculational conditions. If the number of turns of wire is few, the approximation error becomes large. This deteriorates the performance. To avoid this, the number of turns of wire must be increased, and the spacing between the adjacent turns of wire must be decreased. Thus, the target current distributions must be approximated with high accuracy. This is not a realistic solution.
In the case of a narrow-bore (NB) superconducting magnet (SCM), it is desired to set the inside diameter of the magnetic field gradient coils greater than 50% of the inside diameter of the probe cover. However, this cannot be achieved for the following reason. If the former inside diameter is set greater than 50%, the magnetic field leaking to the outside through the outer coil produces a large amount of eddy currents on the cover, thus reducing the rising speed. To avoid this, it is necessary to increase the number of turns of wire forming the coils and to distribute the turns of wire closely, for improving the accuracy of approximation. This will reduce the leakage of the magnetic field. However, this approach is difficult to accomplish because of coil resistance and restrictions on the mechanical dimensions. Furthermore, electric interference between a DC magnetic field gradient coil and the detection coil operating at hundreds of megaherz is large, lowering the Q of the detection coil. Consequently, the sensitivity drops. As a countermeasure, it is conceivable that an RF shield is formed inside the magnetic field gradient coil. However, this shield heavily interferes with the detection coil. This also lowers the Q.